Constraint Qualifications and Regularity Conditions for Nonlinear Constraint Systems
Abstract
Constraint qualifications (CQs) and regularity conditions (RCs) play a fundamental role in establishing global iteration and oracle complexity results for first-order methods on solving functional constrained optimization problems. Various conditions have been proposed and employed in the literature, with their validity established for different classes of applications. However, the use of different conditions across existing studies makes it difficult to directly compare their theoretical results and underlying assumptions. In this report, we systematically review several widely used CQs and RCs and investigate their relationships. In particular, we establish implications among these conditions and identify cases in which the implications are strict, thereby clarifying their relative strengths. Our results provide a unified perspective on existing regularity assumptions and offer guidance for selecting appropriate conditions in the complexity analysis of first-order methods.
Related
- Constraint Qualifications and Gradient Flows for Block Vanishing Constraint Problems
- Distances on Finite Constraint Systems and LICQ Radii in Nonlinear Programming
- Admissibility-Preserving Constraint Lifting for Conditional Deadbeat Control of Discrete-Time Nonlinear Systems
- RUPA: Nonlinear volume consistency, constraint geometry and singular penalty limits in finite elements
- Global Well-posedness and Asymptotic Analysis of a Damped Nonlinear Wave Equation with a Codimension-One Constraint
BibTeX
@misc{zuo2026constraint,
title = {{Constraint Qualifications and Regularity Conditions for Nonlinear Constraint Systems}},
author = {Shiji Zuo and Qihang Lin and Wei Liu and Yangyang Xu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00526},
eprint = {2611.00526},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00526},
}
RIS
TY - UNPB
TI - Constraint Qualifications and Regularity Conditions for Nonlinear Constraint Systems
AU - Shiji Zuo
AU - Qihang Lin
AU - Wei Liu
AU - Yangyang Xu
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00526
UR - https://arxiv.org/abs/2611.00526
AB - Constraint qualifications (CQs) and regularity conditions (RCs) play a fundamental role in establishing global iteration and oracle complexity results for first-order methods on solving functional constrained optimization problems. Various conditions have been proposed and employed in the literature, with their validity established for different classes of applications. However, the use of different conditions across existing studies makes it difficult to directly compare their theoretical results and underlying assumptions. In this report, we systematically review several widely used CQs and RCs and investigate their relationships. In particular, we establish implications among these conditions and identify cases in which the implications are strict, thereby clarifying their relative strengths. Our results provide a unified perspective on existing regularity assumptions and offer guidance for selecting appropriate conditions in the complexity analysis of first-order methods.
ER -